It depends on the agency and the creditors involved, but in many cases, yes, you can keep one card outside the plan.
The standard condition of a DMP is that all enrolled credit card accounts are closed to new purchases. Creditors insist on this because they're reducing your interest rate as a concession, and they don't want you adding new charges while they're giving you a break on the existing balance.
But "enrolled" is the key word. If you have four credit cards and only put three in the DMP, the fourth stays open and active. It won't get the reduced interest rate, but you can use it for genuine emergencies.
Most counseling agencies will discuss this option with you during the enrollment process. Some actively recommend keeping one card out of the plan as an emergency backstop. Others prefer all cards be included so your total interest savings are maximized and you're not tempted to add new debt.
The rules to follow if you keep a card:
Keep the balance as low as possible. Using it for emergencies only means actual emergencies (car broke down, unexpected medical bill), not a sale at your favorite store.
Pay the full balance each month if possible. You're paying full interest on this card, so carrying a balance defeats the purpose.
Don't let the available credit become a reason to overspend. If having the card available leads to $200 in impulse spending each month, the DMP savings on your other cards are being offset.
Some creditors may close your account with them even if you don't include it in the DMP. When creditors see the DMP notation on your credit report, they sometimes proactively reduce your credit limit or close the account as a risk management measure. This isn't guaranteed, but it's possible.
If keeping a card is important to you, ask your counselor about it during the initial session. They'll tell you honestly whether it's feasible with your specific creditors and financial situation.